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Γ_{00} conjecture characterizing Jacobians via base locus of |2Θ|

Establish the Γ_{00} conjecture asserting that a principally polarized indecomposable abelian variety is a Jacobian if and only if the base locus of the Γ_{00} linear subsystem of |2Θ| associated to a nonzero two-torsion point has the prescribed geometric property characterizing Jacobians.

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Background

The Γ_{00} conjecture proposes a geometric criterion inside the linear system |2Θ| tied to a two-torsion point that singles out Jacobians among principally polarized abelian varieties. It complements trisecant-based characterizations by focusing on a base locus condition in a specific linear subsystem.

The authors emphasize that, despite partial progress in a special case, the conjecture remains completely open in general, reflecting the difficulty of capturing the Jacobian locus through higher-order theta geometry at two-torsion points.

References

There is an open conjecture, called the~$\Gamma_{00}$ conjecture , that characterizes Jacobians in terms of the geometry of the base locus of a linear subsystem of $|2\Theta|$ associated to a two-torsion point. The $\Gamma_{00}$ conjecture remains completely open, except for the easiest case .

— Integrable systems approach to the Schottky problem and related questions (2504.20243 - Grushevsky et al., 28 Apr 2025) in Section “Geometry of Theta Divisors”